Question

Which of the following statements is correct:
$I:$ If $x\cos \alpha +y\sin \alpha =P$ is the tangent with respect to standard ellipse then
$P=\sqrt{a^2{\cos }^2\alpha -b^2{\sin }^2\alpha }$
$II:$ The point of contact of the tangent line $4x+y-7=0$ with the ellipse $x^2+3y^2=3$ is $\left(\frac{12}{7},\frac{1}{7}\right)$

• A. Only
• B. Only II
• C. Both I and II
• D. Neither I nor II

Answer 1

Answer: (B)

Only II

$I:x\cos \alpha +y\sin \alpha =P$

$\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$

$\frac{a\cos \alpha }{\cos \theta }=\frac{b\sin \alpha }{\sin \theta }=\frac{P}{1}$

$\cos \theta =\frac{a\cos \alpha }{P},\sin \theta =\frac{b\sin \alpha }{P}$

${\cos }^2\theta +{\sin }^2\theta =1$

$P=\sqrt{a^2{\cos }^2\alpha +b^2{\sin }^2\alpha }$

$II:xh+3yk=3$

$4x+y=7$

$\frac{h}{4}=\frac{3k}{1}=\frac{3}{7}$

$h=\frac{12}{7},k=\frac{1}{7}$