Home
Class 12
MATHS
If the straight line xcosalpha+ysinalpha...

If the straight line `xcosalpha+ysinalpha=p` touches the curve `(x^2)/(a^2)+(y^2)/(b^2)=1` , then prove that `a^2cos^2alpha+b^2sin^2alpha=p^2dot`

Text Solution

Verified by Experts

We know that the line y=mc +c is a tangent to the ellipse
`(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
if `C^(2)=a^(2)m^(2)+b^(2)`
Then comparing the given line `x cos theta+y sin alpha=p` with y=mx +c, we have c=`p//sin alpha,m=-cos alpha//sin alpha`
So, the given line will be a tangent if
`(p^(2))/(sin^(2)alpha)=a^(2)(cos^(2)alpha)/(sin^(2)alpha)+b^(2)`
or `p^(2)=a^(2)cos^(2)alpha+b^(2)sin^(2)alpha)`
Promotional Banner

Similar Questions

Explore conceptually related problems

If the straight line xcosalpha+ysinalpha=p touches the curve x y=a^2, then prove that p^2=4a^2cosalphasinalphadot

If the straight line y= x sin alpha+ a sec alpha be a tangent ot the circle x^(2)+y^(2)=a^(2) , then show that cos^(2)alpha=1

If the straight line y= x sin alpha + a sec alpha is a tangent to the circle x^(2) + y^(2) = a^(2) then-

In A B C , prove that (a-b)^2cos^2C/2+(a+b)^2sin^2C/2=c^2dot

If the line x cos alpha+ y sin alpha=p be a normal to the hyperbola b^(2)x^(2)-a^(2)y^(2)=a^(2)b^(2), show that, p^(2)(a^(2) sec ^(2) alpha- b^(2)"cosec"^(2) alpha)=(a^(2)+b^(2))^(2)

If the straight line y= 4x -5 touches the curve y^(2) = px^(3) + q at (2, 3), then the values of p and q are-

Find the condition that the straight line x cos alpha+ y sin alpha=p is a tangent to the : ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1

The straight line x+ y= sqrt(2) p will touch the hyperbola 4x^(2) - 9y^(2) = 36 if-

If x/a+y/b=2 touches the curve x^n/a^n+y^n/b^n=2 at the point (alpha,beta) then:

If cos^2theta-sin^2 theta = tan^2 alpha , then show that cos^2 alpha -sin^2 alpha = tan^2 theta .