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Joint Equation of Pair of Lines Joint Equation of Pair of Lines Joining the Origin and the Points of Intersection of a Line and a Curve If the line lx + my + n = 0, (n ≠ 0) i.e. the line not passing through origin) cuts the curve ax2 + by2 + 2gx + 2fy + c = 0 at two points A and B, then the joint equation of straight lines passing through A and B and the origin is given by homogenizing the equation of the curve by the equation of the line. i.e. ax2 + 2hxy + by2 + (2gx + 2fy) (kx+my/–n) + c (lx+my/–n)2 = 0. is the equation of the lines OA and OB. Illustration: Prove that the straight lines joining the origin to the points of intersection of the straight line hx + ky = 2hk and the curve (x – k)2 + (y – y)2 = c2 are at right angles if h2 + k2 = c2. Solution: Making the equation of the curve homogeneous with the help of that of the line, we get x2 + y2 –2(kx + hy) (hx+ky/2hk) + (h2 + k2 – c2) (hx+ky/2hk) = 0 or 4h2k2x2 + 4h2k2y2 – 4h2x(hx+ky) – 4h2ky(hx + ky) + (h2 + k2 – c)(h2x2 + k2y2 + 2hxy) = 0. This is the equation of the pair of lines joining the origin to the points of intersection of the given line and the curve. They will be at right angles if Coefficient of x2 + coefficient of y2 = 0 i.e. (h2 + k2) (h2 + k2 – c2) = 0 ⇒ h2 + k2 = c2 (since h2 + k2 ≠ 0). The platform at askIITians is the only group of IITians which provides you free online courses and a good professional advice for IIT JEE, AIEEE and other Engineering Examination preparation. You can also visit askIITians.com to study topics pertaining to the IIT JEE and AIEEE syllabus for free. To read more, Buy study materials of Straight Lines comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here. To know more about the study material of engineering and medical exams, please fill up the form given below:
If the line lx + my + n = 0, (n ≠ 0) i.e. the line not passing through origin) cuts the curve ax2 + by2 + 2gx + 2fy + c = 0 at two points A and B, then the joint equation of straight lines passing through A and B and the origin is given by homogenizing the equation of the curve by the equation of the line. i.e.
ax2 + 2hxy + by2 + (2gx + 2fy) (kx+my/–n) + c (lx+my/–n)2 = 0.
is the equation of the lines OA and OB.
Prove that the straight lines joining the origin to the points of intersection of the straight line hx + ky = 2hk and the curve (x – k)2 + (y – y)2 = c2 are at right angles if h2 + k2 = c2.
Making the equation of the curve homogeneous with the help of that of the line, we get
x2 + y2 –2(kx + hy) (hx+ky/2hk) + (h2 + k2 – c2) (hx+ky/2hk) = 0
or 4h2k2x2 + 4h2k2y2 – 4h2x(hx+ky) – 4h2ky(hx + ky) + (h2 + k2 – c)(h2x2 + k2y2 + 2hxy) = 0.
This is the equation of the pair of lines joining the origin to the points of intersection of the given line and the curve. They will be at right angles if
Coefficient of x2 + coefficient of y2 = 0 i.e.
(h2 + k2) (h2 + k2 – c2) = 0 ⇒ h2 + k2 = c2 (since h2 + k2 ≠ 0).
The platform at askIITians is the only group of IITians which provides you free online courses and a good professional advice for IIT JEE, AIEEE and other Engineering Examination preparation. You can also visit askIITians.com to study topics pertaining to the IIT JEE and AIEEE syllabus for free.
To read more, Buy study materials of Straight Lines comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.
To know more about the study material of engineering and medical exams, please fill up the form given below:
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